Yes. The Higgs is postulated within the framework of the standard model of particle physics, which takes special relativity as one of its assumptions. Special relativity assumes a flat 4D spacetime called "Minkowski spacetime".

The discovery of the Higgs particle (+ non-discovery of other particles + non-discovery of other effects) would confirm that SM + 4-dim GR is 'correct' and 'complete' within the energy range accessable at the LHC.

No, the Higgs is predicted by the SM. But in order to produce that predition - and, more importantly, explain a large number of other experimental observations, the SM has to "parachute in" a quartic 'Mexican hat' Higgs field potential into its Lagrangian. The SM has no deeper justification for this than that it explains the data.

More likely, most of us probably feel, that Higgs potential is the result of some even more fundamental physics at higher energy levels...

Yes. The Higgs is postulated within the framework of the standard model of particle physics, which takes special relativity as one of its assumptions. Special relativity assumes a flat 4D spacetime called "Minkowski spacetime".

Is this why I have seen some discussion on aspects of time travel in relation to the Higgs boson, in the sense that the past, present, and future are 'laid out' or existing in principle? Or, maybe it is better for me to ask: How does time travel relate the Higgs boson discovery, or is this merely pseudo-science speculation and really nothing to do with science at all?

No, the Higgs is predicted by the SM. But in order to produce that predition - and, more importantly, explain a large number of other experimental observations, the SM has to "parachute in" a quartic 'Mexican hat' Higgs field potential into its Lagrangian. The SM has no deeper justification for this than that it explains the data.

More likely, most of us probably feel, that Higgs potential is the result of some even more fundamental physics at higher energy levels...

So if no Higgs boson is found, what happens to the standard model? Will it essentially stay the same with the addition of the new physics at higher energy?